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Ann. Rev. Biophys. Biophys. Chem. 1987.16:45.5-78Copyright ©
1987 by Annual Reviews Inc. All rights reserved
PHYSICAL LIMITS TOSENSATION ANDPERCEPTION
William Bialek1
Institute for Theoretical Physics, University of California,
Santa Barbara,California 93106
CONTENTS
PERSPECTIVES AND OVERVIEW
........................................................................................
455
FOUNDATIONS
................................................................................................................
456Signals and Noise, Detectability and Discriminability
............................................... 456Noise from
Classical Statistical Mechanics
...............................................................
459Noise from Quantum Mechanics
...............................................................................
460
CASE STUDIES: SIMPLE SENSORY TASKS
.............................................................................
462Photon Counting in l~’ision
........................................................................................
462Threshold Signals in the lnner Ear
............................................................................
464Molecule Counting in Chemoreception
......................................................................
467Infrared and Thermal Senses
....................................................................................
469
CASE STUDIES: SYNTH]ESI$ OF mGrmR-ORDER PERCEPTS
.................................................... 471The Missing
Fundamental
.........................................................................................
471Recognizing Ensembles of lmages
.............................................................................
472Hyperacuity and Visual Movement Detection
........................................................... 473
OUTLOOK
......................................................................................................................
474
PERSPECTIVES AND OVERVIEW
One of the most important conceptual developments in modern
physics isthe appreciation of fundamental limits to the reliability
of even the mostprecise measurements. As these ideas were first
explored in the early yearsof this century, several of the pioneers
of the subject turned to a naturalquestion: To what extent do our
sensory systems, which are after all
t Present address: Departments of Physics and Biophysics,
University of California,Berkeley, California 94720.
4550883-9182/87/0610-0455502.00
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456 BIALEK
physical measuring devices, approach the limits to measurement
imposedby the laws of physics?
The idea that the boundaries of perception are set by
fundamentalpl~ysical principles is very attractive. Tests of this
hypothesis requiredseveral developments:
1. Experimental methodology. At the time that physical limits to
per-ception were first proposed, quantitative description of
perceptual per-formance was in its infancy. It was more than forty
years before theconcept of "sensory threshold" was clarified and
translated into prac-tical experiments.
2. Theoretical complexity. Our sense organs are complicated
structures,and powerful theoretical methods are required to
understand the physi-cal limits to measurement in such systems.
3. Biological significance. Suppose we succeed in demonstrating
that aparticular sensory system reaches the relevant physical
limits to itsperformance. What have we learned?
Substantial advances in each of these areas have been made in
the lastdecade. Perhaps the most important has been the realization
that a sensorysystem that reaches the physical limits to its
performance is exceptional.Broad classes of plausible mechanisms
simply cannot reach these limits,and in favorable cases very
specific requirements are placed on the mech-anisms of filtering,
transduction, and amplification within the receptor cell.
The ideas that are relevant for understanding the physical
limits tosensation and perception range from the quantum theory of
measurementto the zoology of animals adapted to different sensory
environments.Obviously it is impossible to do justice to this
complete range of topics ina short review. What I do hope to
communicate is how studies of thephysical limits to sensory
performance have changed our view of thesensory systems, both at
the level of transduction mechanisms in receptorcells and at the
level of the neural mechanisms responsible for processingsensory
information.
FOUNDATIONS
Of major importance in this review is clarity in the use of
phrases suchas "sensory threshold," for behind fuzziness of
language lurk seriousconceptual problems. Here I review the
theoretical background.
Si#nals and Noise, Detectability and Discriminability
It was once believed that each sensory system has a threshold
below whichstimuli generate no percept. Although variants of this
idea continue to
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LIMITS TO SENSATION AND PERCEPTION 457
appear, the notion of a fixed sensory threshold is wrong. The
theory ofsignal detectability (89, 103) provides an alternative
view, which has beensystematically applied in a wide variety of
psychophysical experiments(61). There is also significant
physiological evidence against the classicalthreshold concept. In
the inner ear, for example, receptor potentials areproportional to
sound pressure (or pressure squared) for small-amplitude,pure-tone
stimuli (42, 45, 60). The probability per unit time of
auditorynerve firing is also modulated linearly or quadratically
(71, 72, 90): arbitrarily small stimulus generates a
proportionately small but still non-zero response.
Limits to the detectability of small stimuli are set by noise.
If we aretrying to measure some stimulus x, we usually observe some
variable ywhose average value is (for example) proportional to x
((y) = ~Tx) fluctuates as described by the probability distribution
P(ylx) for y givenany particular x. These distributions are not
abstract mathematical objects;x and y represent physical
quantities, and hence P(y[x) embodies assump-tions about some
underlying physical processes.
The typical sensory threshold experiment is to ask if we can
distinguishbetween x = 0 (no signal) and some x -- Xo # 0; the two
alternatives occurrandomly with probabilities P(x = 0) and P(x =
x0). If we have justobserved some particular y, the probability
that the stimulus was x -- x0is, by Bayes’ theorem,
P(x=xolY)=P(ylxo)P(x=xo)/P(.v), withP(y) = P(ylxo)P(x = xo) +
P(ylO)P(x = 0); the probability is similarly cal-culated for P(x =
0[y). Optimal unbiased discrimination is based onmaximum likelihood
(61): If 2(y) = [P(x = Xoly)/P(x = 01y)] >0, weguess that y was
generated by the signal x = x0, while if)~(_v) < 0 we guessthat
x = 0. The probability of correctly identifying the signal is
thenPc(x = Xo vs x = 0)= S dy P(y[x0)O[2(y)], where O[z] is the
unit function; ®[z > 0] = 1, O[z < 0] = 0.
An example is independent additive Gaussian noise; y = #x + A,
whereA is a random variable chosen from a Gaussian distribution
whose statisticsare independent of x, so that P(ylx) = (2no-2)- 1j2
exp [- (y-gx)2/2tr2]. the stimuli x = 0 and x = x0 are a priori
equally likely, the maximum-likelihood decision rule is just that x
= x0 ify > gXo/2. The probability ofcorrectly identifying x = x0
is P~(x = xo vs x = 0) = (1/2) [1 with the error function ~(z) =
(2/~)’/2 S~ d~ exp (- (z/2). As the signal-to-noise ratio or
"detectability," d’ = #xo/a, approaches zero, Pc --* ½,
whichcorresponds to guessing randomly x = 0 or x = x0. As d" --* oo
we find
Pc ~ 1, so at large signal-to-noise ratios discrimination is
perfect. Unitsignal-to-noise ratio, d’ = 1, corresponds to Pc =
0.76; this provides aconvenient "threshold for reliable
detection."
When x and y are functions of time (100) we write y(t)=
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458 BIALEK
~dt’g(t, t’)x(t’)+A(t). If y(t) is a physical coordinate that
responds toexternal signals x(t), then the response must be causal;
y(t) cannotdepend on x(t’) unless t >t’. It is also reasonable
to assume thatthe response to a particular stimulus does not depend
on the precisearrival time of that stimulus, so #(t, t’) = #(t- t’)
and #(~ < 0) = 0. larly, the physical processes that generate
the noise are not changingin time; any particular noise waveform
A(t-t0) occurs with equalprobability for any time to. The technical
term for this property is"s~ationarity." For a finite time interval
- T/2 < t < T/2 we can writeA(t) = T-~/2 Y.nAnexp (-in~t/T).
A stationary Gaussian noise source characterized by P({A,})=
Z-lexp[-(1/2)Y~A,A ,/S~]; Z is a nor-malization constant. Other
Gaussian distributions are possible, but theycorrespond to
nonstationary noise. For large T the "variances" Sn are acontinuous
function ofo~ = nn/T, S^(~o) = ~ dz e+iO* ( A(t)A(t- ~)), termedthe
spectral density of the noise. The spectral density can also be
definedby averaging the Fourier components2 of A(t): z~(~o)=
~dte+iO*A(t),(~’,(co)~(o9’)) = Sa(co)2n6(co+co’). Noise sources are
often characterizedby qualitative features of their spectra. Thus
approximately contant S(~o)is termed white noise, S(~o) ~,, 1/~o is
"l/f" noise, and so on.
Computation of the probability of correct discrimination
between, forexample, x(t) = 0 and x(t) = Xo(t) is now
straightforward. The result is ofthe same form as before, with an
effective signal-to-noise ratio
f&o 1(d’)2 = 2n Sa(o))[ff(°))~°(~°)12" 1.This result
provides a rigorous basis for comparing sensory thresholdswith
physical models. If the model noise source is Gaussian, d’ = 1
definesa signal level such that 76% correct signal identification
is possible. If wehave correctly identified the noise source it is
impossible for a smallersignal to be detected with this
reliability. Should smaller signals be reliablydetected we have
unambiguous evidence that our physical model is incor-rect.
A second example of signal detection concepts is in modulated
Poissonprocesses (100). These processes consist of discrete events,
such as nerveirnpulses or photon arrivals at the retina,
characterized by their arrivaltimes t,, with the probability of an
event occurring between t and t+dtgiven by r(t)dt. Imagine now that
we are forced to discriminate, in a timeinterval 0 < t < T,
between two signals that give rise to rate functionsr÷(t) and
r_(t). In the limit that the "signal" r+(t)-r_(t)= fir(t) is
:z In the following, J~(~o) always denotes the Fourier transform
of a function f(t), with the"+ i’" sign convention as shown
here.
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LIMITS TO SENSATION AND PERCEPTION 459
small, we can evaluate Pc(+ vs -) in closed form; the result is
identicalto that for a signal in Gaussian noise, with the
detectability(d’)2 = ~rodzrr2(z)/r(z).
Noise from Classical Statistical Mechanics
The basic principle of equilibrium statistical mechanics (85,
88) is that theprobability of being in a configuration with energy
E is proportional toexp(-E/ksT), where T is the absolute
temperature of the environmentwith which the system equilibrates
and Boltzmann’s constant ks, is1.36 × l0 -23 J/K. Because this
description is probabilistic almost anymeasurable quantity will
fluctuate. These fluctuations act as a source ofnoise, which limits
the reliability of any measurement.
For a mass m on a spring of stiffness ~ the energy is E =
½mv2-q-½1~x2,
where v is the velocity and x is the displacement of the object.
ThusP(x, v) = Z- exp (- mv2/2kBT- tcx2/2kBT). We seethatx andv are
inde-pendent Gaussian random variables with zero mean and with
variances((6/-)) 2) = ksT/m and ((rx) 2) = ksT/~c. The mean
potential energy is½x((rx)~) = ½ksT, and the mean kinetic energy is
also ~knT. A system inthermal equilibrium is not a state of minimum
energy, but fluctuatesamong many states; its mean energy is a
measure of the temperature. Thisis the equipartition theorem, and
the fluctuations that enforce this theoremare termed "thermal
noise."
A second example concerns fluctuations in the energy itself. If
the systemhas a set of states with energies E~ the probability
distribution of the energyis P(E) = Z 1Z~6(E-Es)exp(-E~/knT). It
can be readily verified that((6E) 2) = ksTZ(O(E)/OT), where 3(E)/3T
is the specific heat Co. It isconvenient to think of the interal
energy noise as equivalent to temperaturefluctuations: ((6 T) 2)
T~k~/C~.
These results give the probability of finding any particular
coordinateat one instant, If the receptor cell itself does not
filter this coordinate oraverage it over time, then the
equipartition noise level sets the limit to thedetection of small
signals. If the receptor cell does filter the incoming signalwe
need to know the temporal properties of the noise; these are
determinedby the fluctuation-dissipation theorem (FDT).
Consider again a mass-spring system, but now immersed in a
fluid.Interaction with the fluid leads, in the simplest cases, to a
drag forceF~r~g = - v(dx/dt). In addition, random collisions with
the individual mol-ecules of the fluid lead to a fluctuating or
Langevin force 6F. The Langevinequation of motion is then (85)
rn(d:x/dtwhere F~t is an external force. The properties of fiF must
be such that thetotal fluctuations ((6x)~) agree with
equipartition; dissipative processesremove energy from the system
and the fluctuations generate a power flow
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460 BIALEK
into the system to compensate and maintain equipartition. If the
relaxationtime for the energy is ze the power flow should thus be
~k~T/ze. In aresonant system where the response bandwidth, Af, is
,-~ 1/re the "thermalnoise power," P~v, is 4k~TAf (99).
More generally, for a coordinate x that responds linearly to
externalfc.rces ~(o) = 0~(co)ff(co), the response function ~(co) is
in general and its imaginary part 0~"(co) is associated with
dissipation. The FDTstates that x(t) exhibits Gaussian noise of
spectral density (88)S.~(co) = k~Tco-~0~"(co). One can think of
this noise as arising from aLangevin force with spectral
density
=-co I~(o)12- ~ L~(~)J"
In a chemical system, the coordinates are the concentrations of
thevarious species, forces are the free-energy differences among
the species(75), and response functions are determined by the
kinetic equations.Application of the FDT to such models gives
results that agree withstatistical descriptions of molecules
flickering among different chemicalstates (126), but such
statistical pictures require assumptions about thel~,ehavior of
individual molecules beyond those required to reproduce
themacroscopic kinetics. The fluctuation-dissipation theorem
guarantees thatany microscopic mechanism that generates certain
macroscopic behaviornecessarily generates thermal noise of known
characteristics.
Noise from Quantum MechanicsEven at T = 0, in the absence of
thermal noise, most coordinates remainrandom variables. This noise
arises from quantum mechanics and can bequalitatively understood in
terms of Heisenberg’s uncertainty principle,which constrains the
reliability of simultaneous measurements of com-plementary
variables, such as position and momentum. Repeated measure-ments of
these quantities will show that they fluctuate with
standarddeviation fix and 5p; the uncertainty principle states that
5pSx >~ hi2, withPlanck’s constant h = 1.054 × 10-34 J-s. If we
measure the coordinate x(t)as a function of time we can calculate
the velocity and hence the momen-tum, so unless such measurements
have a noise level proportional to h theuncertainty principle will
be violated. The "quantum noise" that enforce~.the uncertainty
principle provides an independent limit to the reliabilityof
measurements.
The spectral density of noise in a quantum system can be
calculated byimagining a sequence of measurements of x(t) at times
t~ (54, 111, 132),
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LIMITS TO SENSATION AND PERCEPTION 461
by considering that the measurement of x(t) constrains the
possible paths(51a) that the system could have taken during the
observation time (38,95). The two methods give identical results
whenever they can be comparedmeaningfully. For example, in a linear
system with response function0~(o~) in equilibrium at temperature
T, the spectral density of coordinatefluctuations is
S~(~o) = ~ ~ (~o) coth (ho3/2kaT)
The first term in Equation 3 is the quantum version of the
fluctuation-dissipation theorem. If the thermal energy kaT is much
larger than thequantum energy hog, we recover the classical result.
This simple energeticcomparison is the same one that governs the
significance of quantum effectsin thermodynamics (88). The second
term in Equation 3 is a dynamicalcontribution to the noise, which
cannot be understood from equilibriumthermodynamics; it expresses
purely quantum phenomena such as"spreading of the wave packet" (51
a). As a result, quantum effects can important even if the quantum
energy h~o is ~knT~"(~o)/l~(~o)], which ismuch smaller than the
thermal energy if the dissipation or damping,~ ~"(~o), is small
(32). This happens in very sharply tuned mechanical electrical
resonators.
There are many situations in which sharp tuning is desirable but
thenatural mechanical or electrical parameters of the system do not
allow it.Under these conditions we can modify the passive dynamics
using activefeedback: We measure some coordinate, amplify and
filter the signal, andapply a force back to the system in
proportion to this signal. If we startwith a system whose response
function is 8(~o) and apply a feedback force
Pr~db~ck(~O) = ~7(~0)~(~0), then a(~o) ~ *7~fr(o~) =
[~-1(o~)--~7(o~)]-~. In thisway we can synthesize resonances at
frequencies where there are no res-onances in the passive system or
sharpen the frequency selectivity of anexisting resonance. We
expect that as the response bandwidth of the activesystem is
narrowed quantum noise becomes more significant.
The dominant noise source in a very narrow-band active
feedbacksystem is the amplifier. Viewed quantum mechanically, an
amplifier is adevice that effects a transformation from a set of
coordinates at its inputto a corresponding set at the output. Any
such transformation that arisesthrough the time evolution of a
physical system must be unitary; it mustpreserve the overall
conservation of probability as well as the formalstructure of
complementary variables on which the uncertainty principleis based.
This requirement results in a quantum limit to the noise level
oflinear amplifiers (37). A feedback system is more complicated
than
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462 BIALEK
isolated amplifier because the output of the amplifier is
connectedb~,ck to the input, but the physics is the same (19). In
our example withffr,.’edba~k(m) = )r(~o)~(a~), the amplifier
contributes an effective noise with spectral density S~(~o) ,-~
½hl~(~o)1218cf~(~o)l. As the bandwidthdecreases, 8o~(~o) becomes
more sharply peaked and the noise forceincreases for frequencies
near resonance.
A different quantum noise is usually associated with optical
phenomena.The energy of a system has contributions both from
coordinates (potentialenergy) and from their conjugate momenta
(kinetic energy). A classicalforce couples only to the coordinate.
Thus we expect that such a forcecannot, by the uncertainty
principle, deposit a definite energy in the oscil-lator. Since the
oscillator energy is in discrete quanta (photons in
theelectromagnetic case) of magnitude hco, a classical force must
produce superposition of states with different numbers of quanta.
In this casethe probability of finding a given number of quanta
obeys the Poissondistribution, although other distributions are
possible (56). This ran-domness of signals, e.g. photon arrivals at
a detector, means that measure-merits of energy or light intensity
are subject to an irreducible quantumnoise.
CASE STUDIES: SIMPLE SENSORY TASKS
I:a this section I examine the simplest tasks that confront the
sensorysystems: detection of small signals and discrimination of
small changes ina constant background. I emphasize not only the
extent of agreementbetween theoretical limits and observed
performance, but also the moreg:eneral lessons about function and
mechanism that can be learned fromthis comparison.
Photon Countin# in Vision
In 1909 Lorentz suggested that the eye could count single
photons.3 In the1.940s de Vries (50) and Rose (105) noted that the
random arrival photons at the retina also sets a limit to the
reliability of intensity dis-crimination. Their prediction that the
threshold for reliable discriminationvaries as diI ,,~ I~/2 has
been confirmed over a range of/in both behavioraland physiological
experiments (5).
At about the same time that de Vries carried out the above work,
Hechtet al (65) and van der Velden (125) independently mounted one
of classic experiments of modem biophysics. Suppose that effective
absorp-
a For a discussion of the history see Bouman (31). I am grateful
to Professor Bouman forclarifying Lorentz’ role in the development
of these ideas. Of further historical interest i~;Bohr’s (30)
discussion of possible quantum limits to other senses.
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LIMITS TO SENSATION AND PERCEPTION 463
tion of k or more photons by the retina constitutes "seeing." If
a flash oflight delivers on average N photons, then as noted above
there are con-ditions under which the number of photons actually
absorbed by the retinawill be a random variable chosen from a
Poisson distribution whose meanis QN, with Q the quantum
efficiency. The probability of the observerseeing is then
P~ : e- Q~ ~~=~ I!
o
The curve Ps vs log N has a shape that is diagnostic of k and
independent
of Q.Hecht et al and van der Velden measured the frequency of
seeing and
compared it with the predictions of Equation 4. Hecht et al
found excellentfits to their data for k --- 5-7 photons; van der
Velden found k = 2 photons.Since the few photons in question are
distributed over several hundredreceptor cells, the probability
that any one cell captures two photons isminiscule. Absorption of a
single photon must thus produce a significantsignal in one
receptor, and each such signal must contribute to vision.
Teich et al (122, 123) performed frequency-of-seeing experiments
withtwo different light sources, one with Poisson statistics and
one for whichthe photo-count distribution is broader than Poisson.
They observeddecreased reliability of perception with the latter
light source, as expectedif the variability of human responses is
indeed set by the random absorptionof photons.
Sakitt (110) asked subjects to score visual stimuli on a scale
from zeroto six based on perceived intensity. The mean rating
varied linearly withthe mean number of photons arriving at the
cornea. If the rating is equalto the number of photo-counts at the
retina then the probability of a ratinggreater than or equal to k
should follow Equation 4; in Sakitt’s experimentthere are five
independent curves that must all be fit by the same value ofQ.
Agreement with the Poisson model is excellent. Apparently humanscan
quite literally count single photons.
Single-photon responses from photorcceptor cells were first
reported inintracellular voltage records from Lirnulus (55).
Voltage recordings fromsingle vertebrate rod cells do not reveal
obvious quantal responses becausethe rods are electrically coupled
(39), but the photocurrent can be directlyrecorded by sucking the
outer segment of the rod into a pipette (8). response to one
photon, a rod cell from the toad, Bufo marinus, producesa current
pulse (9) that can often be fit by I(t)=Io(t/z)3exp(-t/z),with I0 ~
1.2 pA and z ,-~ 0.7-1.4 s. The current noise consists of
twocomponents (10). The first is Gaussian noise with spectral
density
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464 BIALEK
$1(~0) = S~(0)/[1 +(~oz)z]-2; S~(0),-~ 2 Hz-~. The detectability
of asingl¢ photon, from Equation 1, is
fdo~ 1 fo ~ I(t) e+i~t 2 i~0z(d’)2 = 2g S,(-~o) dt - 2S,(0)
30-75.The probability of confusing a single photon with the
Gaussian noisebackground is thus less than 1%. The second noise
arises from spon-taneous current pulses occurring at a rate r =
0.021 s-~ (20°C), which areindistinguishable from the single-photon
response. The level of this noiseis very close to psychophysical
estimates (4) of the "dark noise" level vi’sion (10, 11).
The retinal chromophore of the visual pigment rhodopsin
undergoeseis/trans isomerization in response to photon absorption
(27). Theobserved dark noise places a bound on the rate of
spontaneous iso-merization; with 2 x 109 rhodopsin molecules in the
pipette, the rate permolecule is 10-1~ s-~, or once every 3000
years (10, 11). If the rate forretinal in solution, 4 × 10-8 s-~ at
20°C (68), applied to the actual visualpi.gment, photon counting
would be impossible. In contrast, the photo-isomerization rate is
> 3 × 10~ s-~ (27), and the initial event must be evenfaster to
account for the low quantum yield of fluorescence (51); on
thesetime scales the molecule is not at a definite temperature
(18). The photo-isomerization rate of free retinal is 109 s-~ (70).
Note that fluorescencequenching is functionally important; without
it a photon absorbed at onepoint would be re-emitted and possibly
counted in a neighboring rod cell,so that both sensitivity and
spatial resolution would be sacrificed.
Since the activation energies for spontaneous isomerization of
free reti-nal and rhodopsin are similar (10, 11, 68), the
suppression of dark noiseis not simply an energetic effect but is
dynamical, as are the special featuresof ultrafast reactions (18).
Thus to understand how the rod cell countssingle photons, we
require a clear physical picture of how protein dynamicscan
influence reaction rates (20, 59).
ThreshoM Signals in the Inner Ear
The classic experiments of von Brkrsy (127) and Autrum (2, 3)
suggestedthat hearing organs can respond to displacements of
atornic and subatomicdimension. Widely divergent claims have been
made regarding the relationof these displacements to the expected
physical noise level (3, 50a-c, 52,62, 91). Schweitzer and I
re-examined this issue and reported preliminaryconclusions
beginning in 1980 (16-18, 22-24, 114). New experiments
allowimproved estimates of the threshold signals, and parameters
that we had
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LIMITS TO SENSATION AND PERCEPTION 465
to guess in our original noise calculations have now been
measured. Wehave thus substantially improved our account of the
noise problem (W.Bialek & A. Schweitzer, in preparation); here
I outline the issues.
In the simplest model of hearing the hair cell makes
instantaneous, orbroad-band, measurements on the position of its
stereocilia (69). Broad-band measurements of the hair-cell voltage
will then exhibit noiseequivalent to at least the equipartition
noise of the ciliary bundle,((~x)2) = ksT/x. In no inner-ear organ
has the bundle stiffness been foundto exceed x ~ 10-3 N m-1 (44,
53, 67, 119). Thus 6XRraS > 2 x 10-9 m.
In the turtle (1, 42) the sensitivity of hair-cell voltage to
sinusoidal soundpressure at the eardrum is ,-~ 103 mV Pa-1, the
broad-band noise level is1-5 mV, and the displacement of the
basilar membrane (on which the haircells sit) per unit pressure is
,-~ 85 nm Pa-1 (43). These data determine equivalent broad-band
displacement noise of 8.5-43 × 10- ~ m.
In the guinea pig (45, 107) inner hair cells produce median
responses 0.2 mV to pure-tone sound pressures at the behavioral
threshold; the mostsensitive cells produce 1.6 mV. Noise levels
appear to be less than 1mV. In M6ssbauer studies of the guinea pig
basilar membrane (116) thethreshold for observing the "compound
action potential" correspondedto a displacement of ,-~ 3.5 × 10-1°
m, but this threshold is typically (46)10-20 dB above the
behavioral threshold for reliable detection. In inter-ferometric
studies of the cat cochlea (78-82), displacements of at most10-t° m
were observed at 23-25 dB SPL (dB re 20 #Pa), while the
meanbehavioral threshold at these frequencies is -5 dB SPL at the
eardrum(94). These data suggest that displacements smaller than -1°
m arereliably detected. Further evidence for the detection of
subgtngstr6m signalsis provided by the vibratory receptors of the
frog Leptodactylus albilabris(92).
The signals that can be reliably detected in broad-band
recordings fromsingle hair cells are at least 25 dB smaller than
our estimates of the broad-band ciliary displacement noise. This
suggests that the hair cell in factfilters the mechanical signal.
Other possible solutions, such as amplificationof basilar membrane
motion, suffer from independent noise problems (17,18).
If ciliary mechanics are dominated by stiffness and damping, the
spectraldensity of displacement noise from the FDT is Sx = kBTT[x~
+ (~0)~]-1.
From attempts to displace ciliary bundles with fluid streams
(53), an upperbound on the drag coefficient has been determined
(24): 7 < -8 N-sm-1 in the 1 kHz region. This value is not far
above estimates fromhydrodynamic models (33). With y = -8 N-s m-1
and ~c= 10- 3 N m-1,Sx-=-(6 × 10-1:): 2 Hz-~. Reliable d etection
of d isplacements smallerthan 6x ~ 1 A requires a bandwidth Af<
100 Hz, in agreement with
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466 BIALEK
direct measures of auditory tuning (91). The threshold power,
4k~TAf, is-,~ 1.6 × 10-18 W, in agreement with experiment (83).
If the bundle were passively resonant (129) forces would be
filtered butthe total (equipartition) displacement noise would just
be compressed intoa narrow band around the resonance, not reduced.
Electrical filtering withco nventional membrane channels (126) is
necessarily noisy, and is unlikelyto overcome thermal noise without
additional amplification (18). Theseare not arguments against the
existence of such mechanisms, but ratherindications that they are
not sufficient to solve the noise problem.
One mechanism by which the hair cell can solve the thermal
noiseproblem is active feedback. As noted above, application of a
feedback forceproportional to an amplified and phase-shifted
version of the displacementallows substantial bandwidth narrowing
and hence thermal noisereduction (97). Gold (57) suggested in 1948
that active mechanisms mightoperate in the inner ear, although he
focused on basilar membrane ratherthan stereocilium mechanics. A
biological example of active mechanicalfiltering is provided by the
asynchronous insect flight muscles (133).
Gold noted the most dramatic property of active filters: If they
becameunstable under pathological conditions the ear would emit
sound. Con-sistent observation of spontaneous and evoked acoustic
emissions fromthe ear began in 1978 (76, 77, 91,135), and the
active filter hypothesis wasrevived (47, 48, 77). Narrow-band
acoustic emissions can, however, explained even if active elements
are not present in the mechanics of theinner ear (J. B. Allen,
unpublished; 91).
Analysis of instabilities of active filters in the presence of
noise (18, 23,25) reveals qualitative statistical properties that
cannot be reproduced bypassive, stable systems. If x = 0 is a
stable point, the probability dis-tribution of x has a local
maximum at x = 0; the system tends to spendas much time as possible
at the stable point. The opposite is true for antrustable system;
the system tries to diverge from x = 0, and the distributionhas a
local minimum at this point. The probability distribution for
ear-canal sound pressure in frequency bands surrounding an emission
fromthe human ear (25, 134; W. Bialek & H. P. Wit, in
preparation) shows clear local minimum of the distribution at zero
sound pressure. Otherstatistical properties agree quantitatively
with predictions from simpleactive filter models.
A second approach to testing the active filter scenario is to
search for’violations of the fluctuation-dissipation or
equipartition theorems (16, 18).These theorems apply to a system in
equilibrium with its environment,while an active filter is held
away from equilibrium by the feedback forcesfrom the amplifier.
Direct measurements on ciliary bundles from the turtlebasilar
papilla have demonstrated that equipartition is indeed
violated.
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LIMITS TO SENSATION AND PERCEPTION 467
(44). This is strong evidence that the picture derived from the
noise analysisof active filtering at the level of the stereocilia
is qualitatively correct.
Limits beyond those imposed by thermal noise are imposed by
quantummechanics. Calculation of these limits requires more care4
(W. Bialek A. Schweitzer, in preparation), but to understand why
quantum noise canbe relevant a less thorough approach is
acceptable.
If feedback is applied to generate a resonance at frequency co
in abundle dominated by stiffness and damping, then from the
resultsquoted above the quantum force noise added by the amplifier
is6F~,Ms ’~ (hcox/2)~/2 ~ 10 ~7 N at co ~ 103 Hz. If the ciliary
bundle standsfreely in the fluid, this force noise is equivalent to
a fluid displacementnoise of 6x~ ~ ~SF/~co ,,, 10-~2 m, which is
only about one order of mag-nitude smaller than smallest signals
for which we have evidence of reliabledetection.
At very high frequencies the hair cell responds to a low-pass
filteredversion of x2, the square of ciliary displacement. Such
measurements arequalitatively different from measurements of x
itself and are subject todifferent limits. To see this, note that
the force conjugate to x2 is actuallyhalf the stiffness (~), since
½~x2 is the potential energy. At temperature Tthe mean-square
displacement is just k~T/~, so the response of x~ tochanges in ~ is
described by ~ ~ k~T/~c~. From Equation 3, the effectivequantum
noise level, ~Sx~r, is ~(k~T~/~)TM in an integration time z, oronce
again 6x ~ 10-~2 m for ~ ~ 100 ms (41).
If we could say with certainty that the ear makes a
quantum-limitedmeasurement we could conclude with equal certainty
that the processesof transduction, amplification, and filtering
that make this measurementpossible could not be described in
conventional chemical terms (18, 24).
Molecule Counting in Chemoreception
Many biological processes are chemoreceptive in character. The
truechemical senses, from chemotaxis in bacteria to smell and taste
in primates,are obvious examples, but the immune and hormonal
systems face thesame problems of molecular recognition. Here we are
interested in thosesystems for which the most quantitative data are
available at small signallevels.
Stuiver & de Vries (50d, 120) performed a
frequency-of-smelling exper-iment on human subjects. While the
results are not as clear as for vision, theevidence points to
reliable detection of ~ 50 molecules. Similar experiments
4 Our preliminary discussions were based in part on an incorrect
estimate of the ciliarydamping constant, as noted previously (24).
Below I use the estimate derived from Reference53, which I believe
to be accurate.
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468 BIALEK
h~tve been done on the generation of nerve impulses in insect
olfactoryreceptors (74).
Perhaps the most detailed studies of chemoreception at small
signallevels have been studies of bacterial chemotaxis (14). Berg
& Purcell (15)beautifully clarified the theory of noise in such
measurements. I summarizetl~eir conclusions regarding the physical
limits to chemotactic performance:
1. If an organism the size of a bacterium suddenly stops
propelling itself,inertia will carry it only a few hngstrrms
against the viscosity of thesurrounding water.
2. The flux of molecules to the cell is controlled by the size
of the celland the relevant molecular diffusion constant. No
energetically feasibleamount of movement can improve on diffusive
intake. Bacteria canswim to more favorable environments, but this
requires that they beable to sense their environment.
3. The rate at which cell-surface receptors adsorb molecules
from thesurrounding fluid is also limited by diffusion, but the
total rate is nearlymaximal when only a small fraction of the cell
surface is covered byreceptors.
4. If a small cell swims relatively rapidly its attempts to
measure con-centration differences from front to back are subject
to large artifactsand poor signal-to-noise ratios. To measure
gradients the organismmust, while swimming, compare concentrations
at different times.
5. In comparing concentrations the bacterium must swim straight
for aminimum length of time to guarantee that it takes an
independentsample of the concentration. If the time interval is too
long the com-parison does not work because rotational Brownian
motion will haverandomized the cell trajectory.
6. Random fluctuations in concentration and in the occupancy of
cell-surface receptors provide a significant source of noise in
chemotaxis.This can be overcome only by temporal averaging.
7. As with the diffusive flux, the signal-to-noise ratio for
chemoreceptionsaturates rapidly as the number of receptors
increases.
8. The chemotactic performance of Escherichia coli and
Salmonella typhi-murium approaches the limits imposed by chemical
fluctuations. Thesecells must thus count each molecule that binds
to their surface receptors.
A generalization of the Berg-Purcell arguments in which chemical
fluc-tuations are treated as a form of thermal noise is in complete
agreementwith the points above (W. Bialek, unpublished). In
particular, while morecomplex kinetic schemes for reeeptor-ligand
interaction lead to changes in¯ the noise level, the noise has a
minimum level that is set only by the physics.of diffusion. The use
of thermal noise theory also allows us to explicitl2~
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LIMITS TO SENSATION AND PERCEPTION 669
treat the correlations in occupancy of neighboring receptors and
confirmsthe conclusion that as the number of receptors increases,
the signal-to-noise ratio saturates rapidly.
The noise level defines the minimum integration time, ~, within
which adesired concentration change can reliably be detected. Berg
& Purcell (15)calculated ~ for the concentration changes
detected by bacteria under avariety of conditions and found ̄ ~
0.1-1.5 s. However, the cell becomescompletely disoriented as a
result of rotational Brownian motion in just afew seconds. The
bacterium must thus integrate for ~ ¯ and "forget" ona time scale
of ~2-10 r, thus behaving like a bandpass filter with
centerfrequency of -,~ 1 Hz. Direct measurements of the temporal
response ofbacterial motility to brief pulses of chemoattractants
or repellents havequantitatively confirmed the predicted bandpass
behavior (28).
Infrared and Thermal Senses
In dark caves or the desert night the infrared region of the
electromagneticspectrum may provide more information than the
visible. Several speciesof snakes possess specialized pit organs
that make use of this information.Here I discuss the performance of
this sensory system as well as theremarkable antennal thermosensors
of the cave beetle Speophyes lucidulus.
Two very different sensory mechanisms have been considered: a
pho-tochemical one as in ordinary vision, where the cell counts
photons, anda bolometric one in which the cell detects the
temperature rise that resultsfrom absorption of radiation. In the
bolometric picture the cell’s tem-perature, T, determines the mean
rate at which the cell radiates power tothe world through some
relation P(T). In thermal equilibrium 6T fluc-tuates, and we
assume, following the Langevin approach (96), that
thesefluctuations arise from some fluctuating power 6P with
spectral density Spchosen so that ((rT) z) --- k~T2/Cv. Thus Sp =
k~T2(Op/OT).
If the cell is strongly absorbing (63), the minimum noise level
is set blackbody radiation, for which/~(T) can be written as an
integral overthe Planek distribution,
/" d3k hf~kP(T) 2Ac "j m,) exp l’
where A is the cell area, e is the speed of light, k is the wave
vector of thephoton, and ~k = c[k[. At T = 300 K, Sp = 6 x 10-28
[A/100 (~m)2] Wz
Hz-~; the threshold for reliable detection in a measurement
bandwidthAf is 6P = (2SpAf)1/2. On long time scales, 6P produces
temperaturefluctuations, 6T, of6P(O/~/OT)-~ or ,-~ 10-~ K [(Af/1
Hz) (100 (12m)~/A)]m.
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470 BIALEK
If we describe the cell as a photon counter, the random arrival
of blackbodyphotons provides a microscopic basis for the Langevin
tiP (87). From theFIDT we know that this approach must give the
same results as the bolo-metric calculations.
Experiments on single neurons in crotalid snakes (36) and on the
strikingreaction of these animals (63) suggest thresholds for P of
~4 × 10 9 But these results do not establish the threshold for
reliable detection; thesnake probably strikes only at targets whose
infrared emission approxi-mates that typical of a warm-blooded
creature, while the single-unitthresholds were based on subjective
criteria. When the temperature of thereceptor was varied (36),
fT--0.025 K produced ~50 -~ i ncreases i nfiring rate; the
spontaneous rate was r0 ’~ 10 s-1. The firing is approxi-mately a
Poisson process, and the increase in firing rate is proportional
tothe temperature change, so these two results determine an
equivalenttemperature noise of 6T ,~ 10-3 K (Af/Hz)~/2.
More recent studies in the eyeless cave beetle Speophyes
lucidulus confirmthe reliable detection of millikelvin signals and
illustrate the importanceof sensory ecology in guiding biophysical
experiments (40, 93). Corbirre-q-ichan6 & Loftus (40) measured
thermal signals in the beetle’s naturalhabitat, and sometimes found
temperatures stable to + 0.01 K over severalx~finutes. With
temperature drift of several millidegrees per second theywere able
to correlate the firing rate of the antennal thermoreceptors
withboth the temperature and its time derivative as r( t) ~ r o +
~6 T( t) + flf ~( with typical parameters ~ ~ 7 s-1 K-~ and /~-,~
780 K-~; spontaneousfiring rates, r0, were 5-10 s- ~. Again
assuming that neural firing is a Poissonprocess, we find that in a
1-Hz band surrounding 2 Hz the equivalenttemperature noise of these
thermoreceptor neurons is --- 2 × 10-4 K, whichcompares favorably
with the theoretical limit.
Limits also arise from the transduction mechanism. Consider
threepossibilities:
I. Thermal expansion of some structure may be followed by
mechano-reception. Typical thermal expansion coefficients in
proteins are~ 10-4 K-~ (104), so even if the relevant structure is
as long as thesensory hair (93) it will expand by only ~ 10 " m in
response to thethreshold signal. This is in the same range as the
displacement signalsdetected by the ear, so we have the same noise
problems.
?..Temperature effects on a chemical reaction rate may be
followed bychemoreception. Reaction rates typically vary as R ~ A
exp (-- Ea/knT),where E~ is the activation energy. For reliable
detection of the ratechange in a time ̄ we require fir >~
(R/~)~/2. Thus for ~ ~ 1 s and a fairlygenerous E~ of ~0.25 eV, R
>~ 109 S-~ for 6T~ 10-3 K. But if the
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LIMITS TO SENSATION AND PERCEPTION 471
reaction in question releases a reasonable free energy (,,~ 0.5
cV), thisreaction will dissipate a power comparable to the total
metabolic poweroutput of a cell. Even this is insufficient unless
the cell counts everymolecule.
3. Temperature effects on membrane conductance may be followed
byelectroreception. If the conductance is activated as above, G ~
Aexp(-Ea/kBT), and the driving force is ,-~100 mV, shot noise
andthermal (Johnson) noise both require that the modulated current
~ 100 pA to reliably detect ~T ~ 10-3 1~. This is comparable to
themaximum transduction current in many receptors.
These are not rigorous arguments. The point is that millidegree
tem-perature changes can be detected by biologically plausible
mechanisms,but only if these mechanisms reach their respective
physical limits.
CASE STUDIES: SYNTHESIS OF HIGHER-ORDERPERCEPTS
The reliability of perception is determined in part by the
efficiency andeffective noise level of the computations performed
by the central nervoussystem. In the past decade a few key
experiments, guided by theoreticaldevelopments, have probed this
computational noise level in interestingways.
The Missin9 FundamentalIf we listen to a sound with components
at harmonically related frequenciesnfo, we assign to this sound a
pitch f0 even if the component n = 1 is notpresent. Seebeck (115)
discovered this in the last century, but the issuelanguished until
Schouten’s (112) work nearly one century later. The richvariety of
phenomena associated with the missing fundamental has beenreviewed
by de Boer (48a).
Synthesis of the missing fundamental is now viewed as pattern
rec-ognition in the spectral domain: Given a set of resolved
spectral com-ponents at frequencies fu, the brain discerns pattern
f~ ~ nufo, where nu isa set of integers. But fu is represented in
the nervous system by some noisyestimate xu. This noise can be seen
in simple frequency discriminationexperiments, in which stimuli at
f and f’ are reliably discriminableonly if [f-f’[ >~ al(f),
where a~(f) is an effective Gaussian noise level;al(f~ 1 kHz) ~ 3
Hz for sounds of moderate intensity. If the auditorynervous system
registers the set of estimates {xu}, the probability that thisarose
from a harmonic sequence with fundamental f0 is roughly
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P({nu}, f0) ~ ~ exp - -2 ~u ~,(nufo)--~ J = 2 exp [- E({nu},
f0)].
For the best estimate of f0 we should apply maximum likelihood,
mini-mizing the energy E({nu}, f0) as a function of f0 and the
integers
Goldstein (58) made these ideas precise and noted that the
relevantnoise level o-(f) is an effective noise level at the point
where the energyminimization is performed. Thus the set of
parameters ~(f) must be fit data on pitch matching, discrimination,
and identification. In Goldstein’soriginal estimates, derived
largely from the musical interval recognitionexperiments of Houtsma
& Goldstein (66), a(f) was three to ten timeslarger than
~rl(f). This suggests that the added complexity of the
pitche~:traction computation is associated with a tremendous
increase in theeffective noise level, with the "temperature" ~2(f)
increasing by as muchas a factor of one hundred.
Beerends & Houtsma (12) have remeasured o-(f) by testing
regognition&the familiar notes do, re, mi, fa, so in harmonic
tone complexes missingtheir fundamentals. These authors suggested
that this experiment is muchless contaminated by memory effects
than the earlier study (66), whichrequired recognition of the
chromatic notes F_}’, E, F, F~, G, ete. In thenewer results ~(f) is
much closer to a~(f), perhaps within a factor of which suggests
that the nervous system can perform complex computationswithout
much added noise. A similar conclusion is suggested by the
obser-vation (12) that a(f) determined in the identification of
single pitches be used to predict performance in the more complex
tasks of separatingthe pitches in simultaneous harmonic
complexes.
Recognizing Ensembles of Images
]’he frequency-of-seeing experiment indicated that variability
of humanresponses is controlled by characteristics of the stimulus
and not by thesensory nervous system. This idea can be adapted to
complex perceptualtasks by asking observers to discriminate not
between particular pairs ofstimuli but rather between statistical
ensembles of stimuli. This approachwas first used by Julesz (73),
who tried to identify some qualitative featuresof texture
perception.
Random dot patterns can be generated according to many
different~..tatistieal rules. Barlow & Reeves (6, 7) asked
their subjects to discriminatebetween patterns in which the rule
produced correlations between thepositions of dots on opposite
sides of a symmetry axis and patterns inwhich this correlation was
absent. Since the correlation is never perfect,l:here is a finite
value of d’ that is the optimum discriminability for thatpair of
ensembles. To reach optimum performance the observer must (a)
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L~TS TO S~NSAT~O~ A~rD ~RCEPT~Or~ 473
not add significant noise to the image as it is processed, (b)
make full useof a priori information about the statistical
properties of the ensemble,and (c) reliably compute the relevant
likelihood ratios.
Under certain conditions the performance of human observers in
theBarlow-Reeves experiments corresponded to a value of (d’) z
within a factorof two of the theoretical optimum. As the observer
synthesizes the Gestaltpercept of symmetry he or she makes use of
essentially all the informationin the image; the reliability of the
percept is limited by the statisticalproperties of the image and
not by the computational limitations of thenervous system.
Generalizations of this conclusion may be explored usingensembles
that require the observer to perform ever more complex
com-putations to make full use of the available information
(26).
Hyperacuity and Visual Movement DetectionHuman observers
reliably distinguish between a single line and a pair oflines with
equal total brightness at line separations of ~ 1’ of arc. In
spatialinterval detection, where two parallel lines are separated
by angle 0 or0+60, one finds discrimination is reliable at 60 ~<
6" of arc, which is theangle subtended by two inches viewed from
one mile, or one tenth of thespacing between receptors on the
retina (130). Similar results are obtainedfor vernier acuity or for
the detection of movement across the visual field.This remarkable
performance is termed hyperacuity (131).
Until recently there were no data indicating hyperacuity in the
responseof single neurons. In 1984 de Ruyter et al (49b) reported
preliminary resultson the reliability of coding in a single
wide-field movement-sensitive neuron[HI (64)] in the blowfly visual
system. Movement steps trigger severalspikes, and by presenting
each stimulus ~ 104 times we accumulated goodestimates of the
probability distributions P({tu}lOo) for spike arrival timest,
conditioned on the step size 00. From these distributions we were
ableto compute the probabilities for correct discrimination of 00
vs 0~ usingthe maximum likelihood formalism reviewed above.
Discrimination basedon just two or three spikes following a step
was sufficient to allow detectionof 60 = 100-- 0~1 with ~ 80%
reliability for 60 of approximately one tenththe spacing between
receptor cells on the retina (21, 49a).
In the monkey striate cortex the probability of spike generation
inresponse to gratings is a very steep function of grating position
(102); thesedata suggest that a simple spike/no spike decision rule
allows reliablediscrimination between gratings whose positions
differ by roughly one fifthof a receptor spacing. Similar
conclusions have been reached about vernieracuity tasks in cat
cortical neurons (121), while somewhat less quantitativeevidence
suggests performance in the hyperacuity range for cat
retinalganglion cells (117).
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474 BIALEK
Robust intracellular recordings can also be taken from the
photo-receptor cells of the fly. Such recordings have indicated
that the cellvc.ltage responds linearly to the contrast pattern
C(4~, 0, with independentGaussian additive noise:
V.(t)=fd~T(~)fd4~M(4~-4~o)C(4~,t-z)+,V.(t).
The nth receptor cell is located at position ~b, and T and M are
temporaland spatial transfer functions, respectively. For
simplicity I indicate onlyone angular coordinate; 6 V, is
characterized by the spectral density Sv(~).All of these quantities
have been directly measured (49a), and within thismodel the limit
to discrimination between different trajectories, O(t), canbe’,
calculated for a given pattern. The optimal (d’) 2 for step-size
dis-erimination is within a factor of two of the observed neural
discriminationperformance during the behaviorally relevant (86) 40
ms after the step(49a). Apparently a few spikes carry essentially
all of the informationabout movement that is available at the
retina; the fly’s visual systemperforms an optimal and essentially
noiseless processing of the photo-receptor voltages.
G, UTLOOK
In his classic book What is LifeL Schr6dinger (113) drew
attention to theremarkable precision achieved by the cellular
mechanisms responsible fortransmission of genetic information.
Schr6dinger introduced the "naiveclassical physicist" who, not
knowing quantum mechanics, does notappreciate that one can
construct tremendously stable structures on themolecular scale. In
the absence of such "micro-stability," this scientist canunderstand
the stability of the genome only in macroscopic, statisticalterms,
as a collective property of large numbers of molecules. While
thisappears to be a reasonable point of view, it turns out to be
wrong; geneticinformation is stored in single molecules.
It has conventionally been assumed that the nervous system’s
ability toperform reliable computations is similarly a macroscopic
phenomenon,the result of averaging over the erratic behavior of
many individual neuralelements? Against this view, Delbrfick (49)
suggested that the sensorysystems may yet reveal a precision and
subtlety of mechanism far beyondthat appreciated from molecular
genetics. Bullock (34, 35) has cautioned
SThis view is often traced to von Neumann, but from the text of
his Silliman Lectures(128) I think the attribution may be
inaccurate.
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LIMITS TO SENSATION AND PERCEPTION 475
that evidence for imprecision and unreliability in the nervous
system mayequally well reflect our ignorance of how information is
encoded.
The case studies reviewed above provide examples of single
receptorcells that perform their appointed tasks with a precision
and reliabilitythat approaches the limits imposed by the laws of
physics. This fact aloneallows the prediction of qualitatively
significant cellular mechanisms, suchas active filtering in the
inner ear and the integration and adaption pro-cesses of
chemotaxis, which are in fact observed. In at least one
instance--vision--we can approach the problem of sensory
performance at the mol-ecular level, where once again single
molecules are apparently responsiblefor the reliability and
precision of biological function. This example alsobrings us up
against fundamental issues regarding the dynamics of bio-logical
macromolecules. We have seen systems in which complex
neuralprocessing of sensory information can occur with essentially
no additionof noise, where the nervous system succeeds in
extracting nearly all of theavailable information about particular
features of the stimulus, and wherethe results of this optimal
information processing are encoded in just afew spikes from one
neuron.
The evidence for optimal performance of the sensory systems is
stillscattered. It is thus impossible to know whether this
precisionist view ofsensory systems is more generally applicable.
It is easy to formulate criteriafor optimality that are not met in
these systems, but deeper analysis mayreveal that these criteria
themselves are flawed; a nice example is the issueof
diffraction-limited optics in the compound eye (118). In the next
fewyears the approach to optimality should be tested more
quantitatively inseveral systems where the data are already
suggestive.
A more important question is whether the notion of physically
limitedperformance will continue to be a fruitful source of ideas
about functionand mechanism. In this respect the most exciting, if
speculative, possibilityis that these concepts can be applied to a
wider variety of"almost sensory"phenomena. Are the exquisite
spatio-temporal patterns of coordinatedciliary beating (84, 101)
limited in their stability and precision by theBrownian noise
forces acting on the cilia? Is the precise control of polymerlength
and cross-linking observed in stereocilia (124) and flagella
(106)limited by chemical fluctuations during assembly? Are there
fundamentalphysical limits to the precision of molecular and
cellular recognitionevents? Having planted myself firmly on terra
incognita, it is perhaps bestto close.
ACKNOWLEDGMENTS
It is now ten years since Allan Schweitzer and I began to think
about noise
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476 BIALEK
in the sensory systems, and I thank him most of all for our
early discussions.Many colleagues have taught me about issues
relevant to this article; Ithank especially my collaborators R.
Goldstein, S. Kivelson, R. de Ruyter,H. Wit, and A. Zee. I also
thank E. de Boer, H. Duifhuis, A. Houtsma,and J. Kuiper (among many
others), who introduced me to the Dutchtradition in sensory
biophysics. J. Ashmore, M. Goldring, and W. G.Owen will note that
much of what they taught me has not found its wayinto the text;
rest assured that this was only for lack of space. E. Masonachieved
optimal performance in turning my scribbles into text. Work onthis
review was supported by the National Science Foundation underGrant
No. PHY82-17853, supplemented by NASA.
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